Consider a tortoise and hare ambling along a circular track holding the ends of a uniformly stretchable bungee cord between them. When viewing the bungee cord in time and space, the cords form an envelope of a curve. Should the relative speeds of the tortoise and hare remain constant, these curves are the well-known epicycloids and hypocycloids of Euclidean geometry. But what happens in hyperbolic geometry within the unit disk? The result is like visiting a county fair’s hall of curved mirrors. We generate the curves both mathematically and with embroidery hoops, yarn, and weed-whacking string.